TY - JOUR
T1 - Absolute stability and absolute hyperbolicity in systems with discrete time-delays
AU - Yanchuk, Serhiy
AU - Wolfrum, Matthias
AU - Pereira, Tiago
AU - Turaev, Dmitry
N1 - Funding Information:
SY was supported by the German Science Foundation ( Deutsche Forschungsgemeinschaft , DFG) [project No. 11803875 ]. TP was supported by a Newton Advanced Fellowship of the Royal Society NAF Image 1 R1 Image 1 180236 , by Serrapilheira Institute (Grant No. Serra-1709-16124 ), and FAPESP (grant 2013/07375-0 ). MW was supported by the German Science Foundation ( Deutsche Forschungsgemeinschaft ) [project No. 163436311 - SFB 910 ].
Publisher Copyright:
© 2022 Elsevier Inc.
PY - 2022/5/5
Y1 - 2022/5/5
N2 - An equilibrium of a delay differential equation (DDE) is absolutely stable, if it is locally asymptotically stable for all delays. We present criteria for absolute stability of DDEs with discrete time-delays. In the case of a single delay, the absolute stability is shown to be equivalent to asymptotic stability for sufficiently large delays. Similarly, for multiple delays, the absolute stability is equivalent to asymptotic stability for hierarchically large delays. Additionally, we give necessary and sufficient conditions for a linear DDE to be hyperbolic for all delays. The latter conditions are crucial for determining whether a system can have stabilizing or destabilizing bifurcations by varying time delays.
AB - An equilibrium of a delay differential equation (DDE) is absolutely stable, if it is locally asymptotically stable for all delays. We present criteria for absolute stability of DDEs with discrete time-delays. In the case of a single delay, the absolute stability is shown to be equivalent to asymptotic stability for sufficiently large delays. Similarly, for multiple delays, the absolute stability is equivalent to asymptotic stability for hierarchically large delays. Additionally, we give necessary and sufficient conditions for a linear DDE to be hyperbolic for all delays. The latter conditions are crucial for determining whether a system can have stabilizing or destabilizing bifurcations by varying time delays.
KW - Absolute hyperbolicity
KW - Absolute stability
KW - Delay differential equations
UR - http://www.scopus.com/inward/record.url?scp=85125278576&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2022.02.026
DO - 10.1016/j.jde.2022.02.026
M3 - Article
AN - SCOPUS:85125278576
VL - 318
SP - 323
EP - 343
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
ER -